The Tangent Function

Slope on the Unit Circle What is the slope of the terminal side of an angle on the unit circle? 1 (cosӨ,sinӨ) sinӨ Opposite Ө -1 cosӨ 1 Adjacent Using our knowledge of the Unit Circle… Or using trigonometry… Slope = -1

A Definition of Tangent The tangent function is defined as: There are values for which the tangent function are undefined: Any Θ that makes cos(Θ)=0.

Example Find the exact value of the following: Thought process The only thing required for a correct answer (unless the question says explain)

The Tangent Function Graph X SIN(X) COS(X) -2π 1 -7π/4 0.707 -3π/2 -5π/4 -0.707 -π -1 -3π/4 -π/2 -π/4 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π In order to investigate the tangent function, first examine all the values of sine and cosine. Remember, tangent is sine divided by cosine. Now find and graph all of the values of sine÷cosine.

The Tangent Function Graph X SIN(X) COS(X) -2π 1 -7π/4 0.707 -3π/2 -5π/4 -0.707 -π -1 -3π/4 -π/2 -π/4 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π TAN(X) 0/1 = 0 .707/.707 = 1 1/0 = DNE .707/-.707 = -1 0/-1 = 0 -.707/-.707 = 1 -1/0 = DNE -.707/.707 = -1 Find the values of sine divided by cosine. Plot the points. The errors are asymptotes.

The Tangent Function Graph Domain: Range: Asymptotes All Reals except All Reals

Graph of Tangent (For 0 ≤ x ≤ 2π)